GIDE/OGIDE: A Grand Unified Theory of Provably Safe Multi-Agent Interaction Dynamics

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This document presents the formal synthesis of the Layer 5 Global/Capstone Theorems for the Guaranteed Intelligent Dynamics Engine (GIDE) and the Offline General Interaction Design Engine (OGIDE). Structured as a formal preprint, this content integrates individual system completeness, joint integration, and the Grand Unified Theorem (GUT) with the mathematical rigor required for formal verification and defense.

GIDE/OGIDE: A Grand Unified Theory of Provably Safe Multi-Agent Interaction Dynamics
Authors: DarcStar Technologies Research

Document ID: GIDE-PREPRINT-2026-L5

Version: 1.0

Classification: Layer 5 (Capstone)

Abstract
We present the Grand Unified Theorem (GUT) for the GIDE/OGIDE architecture, a quad-domain hybrid dynamical system designed for modeling and controlling complex multi-agent social and physical interactions. By chaining offline dynamics identification (OGIDE) with online adaptive execution (GIDE), we prove that safety invariants are preserved under additive error accumulation. We establish explicit bounds for sample complexity, real-time implementation stability, and resilient recovery after transient assumption violations.

1. Introduction and Foundations (Layer 0) The architecture operates on a compact state space Z⊂R^n and an admissible control space U⊂R^m. The core dynamics f are assumed to be Lf-Lipschitz (Assumption Joint.A4).

1.1 The Configuration Tuple
The bridge between offline design and online execution is the OGIDE output tuple C:C=(θoff, Hoff, Soff, δoff, πoff, Toff)

Where θ represents parameters, H the structure, S the safe set, δ the error budget, π the policy, and T the timescale decomposition.

1. System Completeness (Part I) Theorem 2.1: OGIDE Offline Completeness (OGIDE.Global.T1) The OGIDE system provides a complete offline certification framework characterized by seven primary guarantees:

2.1. Sample Complexity: N = O(M⋅Lθ^2⋅log(∣Z∣/δ)/ϵ^2) ensures ϵ-accurate dynamics fitting.
2.2. Minimax Optimality: The estimator is rate-optimal for the class of Lf-Lipschitz functions.

2.3. Safety Certification: Control Barrier Function (CBF) verification ensures n(z) ⋅ f(z) ≤ 0 for all z∈∂S.

2.4. Transferability: Domain adaptation error is bounded by δ
off+LR⋅ΔP.

Theorem 2.2: GIDE Online Completeness (GIDE.Global.T1)
The GIDE runtime provides an eight-guarantee framework for real-time dynamics:

1. Forward Invariance: Subsystems (Physics, Chemistry, Biology, Information) maintain Z via homeostatic restoring forces.

2. Small-Gain Stability: Stability is guaranteed if the spectral radius of the coupling matrix ρ(Γ) < 1.

3. CfC Stability: The Closed-form Continuous-time update rule is stable for Δt < 2/ωmax.

4. Safe Learning: Euclidean projection onto Θstable ensures adaptation never crosses a bifurcation boundary.

5. Cross-System Integration (Part II)
   Theorem 3.1: The System Integration Bound (Joint.Global.T1)
   The integrated pipeline provides a total end-to-end error bound defined as the sum of phase-specific errors:

δtotal ≤ δfit + δopt + δcert + δtransfer + δadapt
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Safety Margin Preservation: The end-to-end safety margin ηtotal remains positive if the offline margin ηoff exceeds the accumulated transfer and adaptation error:

ηtotal = ηoff − δtransfer − δadapt >0

Theorem 3.2: Computational Complexity Ceiling (Joint.Global.T3)
Safe operation is constrained by a fundamental Compute-Update-Safety triangle:

ηtotal ≤ ηoff − (Lh⋅Csafe⋅Cstep) / Bcompute
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This ceiling establishes that insufficient compute budget (Bcompute) directly necessitates safety margin degradation to maintain real-time deadlines.

1. The Grand Unified Theorem (Part III) Theorem 4.1: Grand Unified Theorem (Joint.Global.GUT) Under the standing assumptions of Layer 0, the GIDE/OGIDE architecture is a sound, complete, and compositional framework possessing eight fundamental properties:

4.1. Quantifiable Error: Every transition in the pipeline has explicit, polynomial-time verifiable bounds.

4.2. Additive Accumulation: Due to contraction mapping, errors sum linearly (∑δi) rather than multiplying.

4.3. Compositionality: Modular small-gain ρ(Γ)<1 permits parallel offline fitting and parallel online verification.

4.4. Resilient Recovery: The system is "self-healing"; it can recover from transient assumption violations in time Trecover = O(ln(δ/ϵ)/λ).

4.5. Semantic Soundness: Interpretation error is bounded and grounded in observable states, preventing semantic "hallucinations".

1. Extended Guarantees (Part IV) Theorem 5.1: Recovery After Violation (Joint.Global.T9) If assumptions are violated for duration Tviolation, recovery is guaranteed if Tviolation <ηreserve/(Lh⋅vviolation). The post-recovery margin ηpost accounts for the safety budget consumed during the "unguaranteed" drift phase.

Theorem 5.2: Verification Decidability (Joint.Global.T11)
Verification complexity is optimized across the stack:

Offline: O(poly(n,d)) for polytopic sets via SDP/SOS relaxation.

Online: O(1) constant-time via the Anytime Safety Certificate:

ηrem(t) = ηoff −δtr −Lh (vdrift⋅t/αPE + (σ/αPE) (1−e^(−αPEt)))

1. Conclusion The GIDE/OGIDE framework establishes a new standard for high-integrity autonomous control. By leveraging the Additive Accumulation Property and Small-Gain Universality, we provide a mathematically closed pipeline that guarantees safety from raw sensor data through to complex semantic interpretation.